scipy tutorial - Baustatik-Info-Server

SciPy Reference Guide, Release 0.8.dev
The first four algorithms are unconstrained minimization algorithms (fmin: Nelder-Mead simplex, fmin_bfgs:
BFGS, fmin_ncg: Newton Conjugate Gradient, and leastsq: Levenburg-Marquardt). The last algorithm actually
finds the roots of a general function of possibly many variables. It is included in the optimization package because at
the (non-boundary) extreme points of a function, the gradient is equal to zero.
1.5.1 Nelder-Mead Simplex algorithm (fmin)
The simplex algorithm is probably the simplest way to minimize a fairly well-behaved function. The simplex algorithm
requires only function evaluations and is a good choice for simple minimization problems. However, because it does
not use any gradient evaluations, it may take longer to find the minimum. To demonstrate the minimization function
consider the problem of minimizing the Rosenbrock function of N variables:
f (x) =
N−1 �
i=1
100 � xi − x 2 �2 i−1 + (1 − xi−1) 2 .
The minimum value of this function is 0 which is achieved when xi = 1. This minimum can be found using the fmin
routine as shown in the example below:
>>> from scipy.optimize import fmin
>>> def rosen(x):
... """The Rosenbrock function"""
... return sum(100.0*(x[1:]-x[:-1]**2.0)**2.0 + (1-x[:-1])**2.0)
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = fmin(rosen, x0, xtol=1e-8)
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 339
Function evaluations: 571
>>> print xopt
[ 1. 1. 1. 1. 1.]
Another optimization algorithm that needs only function calls to find the minimum is Powell’s method available as
fmin_powell.
1.5.2 Broyden-Fletcher-Goldfarb-Shanno algorithm (fmin_bfgs)
In order to converge more quickly to the solution, this routine uses the gradient of the objective function. If the gradient
is not given by the user, then it is estimated using first-differences. The Broyden-Fletcher-Goldfarb-Shanno (BFGS)
method typically requires fewer function calls than the simplex algorithm even when the gradient must be estimated.
To demonstrate this algorithm, the Rosenbrock function is again used. The gradient of the Rosenbrock function is the
vector:
N�
∂f
= 200
∂xj i=1
� xi − x 2 �
i−1 (δi,j − 2xi−1δi−1,j) − 2 (1 − xi−1) δi−1,j.
�
− 400xj
= 200 � xj − x 2 j−1
This expression is valid for the interior derivatives. Special cases are
∂f
∂x0
∂f
∂xN−1
�
xj+1 − x 2� j − 2 (1 − xj) .
�
= −400x0 x1 − x 2� 0 − 2 (1 − x0) ,
= 200 � xN−1 − x 2 �
N−2 .
16 Chapter 1. SciPy Tutorial